In a topological space $(X, \tau)$ a subset $A \subseteq x$ is called: i) pre-open if $A subseteq \operatorname{int}(c 1(A)))^{\prime \prime} A \subseteq A^{-\circ}" .$ ii) semi-open if $A subseteq c 1(\operatorname{int}(A)))^{\prime \prime} A subseteq A^{\circ-"}$. iii) $\alpha-$ open if $A subseteq \operatorname{int}(c 1(\operatorname{int}(A)))^{\prime prime} A subseteq A^{\circ-\circ} "$. iv) $\beta-$ open if $A \subseteq \operatorname{cl}(\operatorname{int} (c 1(A)))^{\prime \prime} A subseteq A^{-\circ-"}$. Show that: a) the family of pre-open sets is closed under arbitrary union. b) the family of $\alpha$-open sets equals the intersection of the families of pre-open and semi-open sets. c) the family of $\beta$-open sets equals the union of the families of pre-open and semi-open sets. CS.VS. 1426||